The Stabilized Supralinear Network is a model of recurrently connected excitatory (E) and inhibitory (I) neurons with a supralinear input-output relation. It can explain cortical computations such as response normalization and inhibitory stabilization. However, the network's connectivity is designed by hand, based on experimental measurements. How the recurrent synaptic weights can be learned from the sensory input statistics in a biologically plausible way is unknown. Earlier theoretical work on plasticity focused on single neurons and the balance of excitation and inhibition but did not consider the simultaneous plasticity of recurrent synapses and the formation of receptive fields. Here we present a recurrent E-I network model where all synaptic connections are simultaneously plastic, and E neurons self-stabilize by recruiting co-tuned inhibition. Motivated by experimental results, we employ a local Hebbian plasticity rule with multiplicative normalization for E and I synapses. We develop a theoretical framework that explains how plasticity enables inhibition balanced excitatory receptive fields that match experimental results. We show analytically that sufficiently strong inhibition allows neurons' receptive fields to decorrelate and distribute themselves across the stimulus space. For strong recurrent excitation, the network becomes stabilized by inhibition, which prevents unconstrained self-excitation. In this regime, external inputs integrate sublinearly. As in the Stabilized Supralinear Network, this results in response normalization and winner-takes-all dynamics: when two competing stimuli are presented, the network response is dominated by the stronger stimulus while the weaker stimulus is suppressed. In summary, we present a biologically plausible theoretical framework to model plasticity in fully plastic recurrent E-I networks. While the connectivity is derived from the sensory input statistics, the circuit performs meaningful computations. Our work provides a mathematical framework of plasticity in recurrent networks, which has previously only been studied numerically and can serve as the basis for a new generation of brain-inspired unsupervised machine learning algorithms.